The Legendre differential equation may be solved using the standard power series method. The equation has regular singular points at x = ±1 so, in general, a series solution about the origin will only converge for |x| < 1. When n is an integer, the solution Pn(x) that is regular at x = 1 is also regular at x = −1, and the series for this solution terminates (i.e. it is a polynomial).

Expanding the Taylor series in Equation (2) for the first two terms gives

P0(x)=1,P1(x)=x{\displaystyle P_{0}(x)=1,\quad P_{1}(x)=x}

for the first two Legendre Polynomials. To obtain further terms without resorting to direct expansion of the Taylor series, equation (2) is differentiated with respect to t on both sides and rearranged to obtain

where r{\displaystyle r} and r′{\displaystyle r'} are the lengths of the vectors x{\displaystyle \mathbf {x} } and x′{\displaystyle \mathbf {x} ^{\prime }} respectively and γ{\displaystyle \gamma } is the angle between those two vectors. The series converges when r>r′{\displaystyle r>r'}. The expression gives the gravitational potential associated to a point mass or the Coulomb potential associated to a point charge. The expansion using Legendre polynomials might be useful, for instance, when integrating this expression over a continuous mass or charge distribution.

Legendre polynomials occur in the solution of Laplace's equation of the static potential, ∇2Φ(x)=0{\displaystyle \nabla ^{2}\Phi (\mathbf {x} )=0}, in a charge-free region of space, using the method of separation of variables, where the boundary conditions have axial symmetry (no dependence on an azimuthal angle). Where z^{\displaystyle {\widehat {\mathbf {z} }}} is the axis of symmetry and θ{\displaystyle \theta } is the angle between the position of the observer and the z^{\displaystyle {\widehat {\mathbf {z} }}} axis (the zenith angle), the solution for the potential will be

where we have defined η = a/r < 1 and x = cos θ. This expansion is used to develop the normal multipole expansion.

Conversely, if the radius r of the observation point P is smaller than a, the potential may still be expanded in the Legendre polynomials as above, but with a and r exchanged. This expansion is the basis of interior multipole expansion.

The trigonometric functions cos⁡nθ{\displaystyle \cos n\theta }, also denoted as the Chebyshev polynomialsTn(cos⁡θ)≡cos⁡nθ{\displaystyle T_{n}(\cos \theta )\equiv \cos n\theta }, can also be multipole expanded by the Legendre polynomials Pn(cos⁡θ){\displaystyle P_{n}(\cos \theta )}. The first several orders are as follows:

Since the differential equation and the orthogonality property are independent of scaling, the Legendre polynomials' definitions are "standardized" (sometimes called "normalization", but note that the actual norm is not unity) by being scaled so that

Pn(1)=1.{\displaystyle P_{n}(1)=1.\,}

The derivative at the end point is given by

Pn′(1)=n(n+1)2.{\displaystyle P_{n}'(1)={\frac {n(n+1)}{2}}.\,}

As discussed above, the Legendre polynomials obey the three term recurrence relation known as Bonnet’s recursion formula

The shifted Legendre polynomials are defined as Pn~(x)=Pn(2x−1){\displaystyle {\tilde {P_{n}}}(x)=P_{n}(2x-1)}. Here the "shifting" function x↦2x−1{\displaystyle x\mapsto 2x-1} (in fact, it is an affine transformation) is chosen such that it bijectively maps the interval [0, 1] to the interval [−1, 1], implying that the polynomials Pn~(x){\displaystyle {\tilde {P_{n}}}(x)} are orthogonal on [0, 1]:

Legendre functions of the second kind (Qn){\displaystyle (Q_{n})}[edit]

As well as polynomial solutions, the Legendre equation has non-polynomial solutions represented by infinite series. These are the Legendre functions of the second kind, denoted by Qn(x){\displaystyle Q_{n}(x)}.

Legendre functions of fractional degree exist and follow from insertion of fractional derivatives as defined by fractional calculus and non-integer factorials (defined by the gamma function) into the Rodrigues' formula. The resulting functions continue to satisfy the Legendre differential equation throughout (−1,1), but are no longer regular at the endpoints. The fractional degree Legendre function Pn agrees with the associated Legendre polynomialP0n.